试题分析:(1)由条件可知,数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207532491.png)
为等差数列,又知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207548435.png)
,其通项公式易求,再根根据数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207516481.png)
与数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207532491.png)
的关系
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207532643.png)
,可求出数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207516481.png)
的通项公式;(2)由(1)中所求的数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207516481.png)
的通项公式,可对
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207657765.png)
进行化简,然后再对其考察;(3)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207688421.png)
时,结合(1)的结果,可求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207953555.png)
,代入
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240242077351022.png)
中,设法对其变形处理,找到
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207984431.png)
的递推关系再进行判断.
试题解析:
(1)因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207563753.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208109750.png)
,所以数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207532491.png)
是以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208140513.png)
为公差的等差数列,又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207548435.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240242081561174.png)
, 2分
故由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207532643.png)
,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208187979.png)
. 4分
(2)因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208203563.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208218666.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208250727.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208265987.png)
,
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208265364.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208281164.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208296387.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208312647.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208328221.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208296387.png)
, 6分
(ⅰ)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208359473.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208374537.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208374337.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208390617.png)
,不符合题意; 7分
(ⅱ)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208406399.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208374537.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208437352.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208452427.png)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208452542.png)
. 8分
综上所述,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208406399.png)
时,存在正整数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207626399.png)
使得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207657765.png)
恒成立,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207626399.png)
的最小值为4.
9分
(3)因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207688421.png)
,由(1)得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207953555.png)
,
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240242086081413.png)
①,
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240242086241597.png)
②,
由②
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208640165.png)
①,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208655755.png)
③, 12分
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208671830.png)
④,
再由④
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208640165.png)
③,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208702569.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208718973.png)
,
所以当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208733435.png)
时,数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207704450.png)
成等比数列, 15分
又由①式,可得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208764395.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208780402.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208811510.png)
,所以数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024207704450.png)
一定是等比数列,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024208842460.png)
.
16分
(说明:若第(3)小题学生由前几项猜出等比数列,再代回验证的,扣3分)